Differential geometry MOC

Differential pullback

Let 𝜑 :𝑀 →𝑁 be a 𝐶𝛼-map. The pullback 𝜑∗ is an operation for “pulling back” data defined on 𝑁 to data defined on 𝑀. Usually this corresponds to some kind of precomposition in the sense of Pushforward and pullback of morphisms.

Differential pullback of a scalar field

Let 𝑓 ∈𝐶𝛼(𝑁) be a scalar field on 𝑁. The pullback 𝜑∗𝑓 ∈𝐶𝛼(𝑀) is defined by diff

(𝜑∗𝑓)(𝑝):=𝑓(𝜑(𝑝))

for 𝑝 ∈𝑀, i.e. 𝜑∗𝑓 =𝑓 ∘𝜑.

Differential pullback of a covariant tensor field

The above may be viewed as a special case of the following. Let 𝜔 ∈T0𝑝(𝑁) be a totally covariant tensor field. The pullback 𝜑∗𝜔 ∈T0𝑝(𝑀) is defined by diff

(𝜑∗𝜔)𝑎1⋯𝑎𝑝(𝑣1)𝑎1⋯(𝑣𝑝)𝑎𝑝:=𝜔𝑎1⋯𝑎𝑝(𝜑∗𝑣1)𝑎1⋯(𝜑∗𝑣𝑝)𝑎𝑝

for vector fields (𝑣1)𝑎,…,(𝑣𝑝)𝑎 ∈𝔛(𝑀), where 𝜑∗ denotes the Differential pushforward of a vector field.

For mixed tensor fields it is in general not possible to define the pushforward, except for the special case of the Differential pullback along a diffeomorphism.

Related

• Tangent map

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